Parabolic induction and restriction functors for rational Cherednik algebras

By R. Bezrukavnikov, Pavel Etingof. 2008

159 citations recorded by the source corpus.

Abstract.We introduce parabolic induction and restriction functors for rational Cherednik algebras, and study their basic properties. Then we discuss applications of these functors to representation theory of rational Cherednik algebras. In particular, we prove the Gordon–Stafford theorem about Morita equivalence of the rational Cherednik algebra for type A and its spherical subalgebra, without the assumption that c is not a half-integer, which was required up to now. Also, we classify representations from category $${\mathcal{O}}$$ over the rational Cherednik algebras of type A which do not contain an Sn-invariant vector, and confirm a conjecture of Okounkov and the first author on the number of such representations. We also prove that the spherical Cherednik algebra of type A is simple for − 1 < c < 0. Finally, in an appendix by the second author, we determine the reducibility loci of the polynomial representation of the trigonometric Cherednik algebra.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/753318a3959335f13e46fc7ef43c330da0bb4ebd

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In Algebra and Number Theory

1. I. Gordon
2. I. Losev
3. E. Gorsky
4. Stephen Griffeth
5. Pavel Etingof
6. D. Hernandez
7. Raphaël Rouquier
8. Ben Webster

In Geometry and Topology

1. I. Gordon
2. I. Losev
3. Stephen Griffeth
4. Pavel Etingof
5. E. Gorsky
6. E. Vasserot
7. D. Hernandez
8. Ben Webster

In Mathematical Physics

1. I. Gordon
2. Stephen Griffeth
3. Pavel Etingof
4. E. Gorsky
5. I. Losev
6. E. Vasserot
7. Charles F. Dunkl
8. Raphaël Rouquier

Citations in the corpus

• Symplectic reflection algebras (2007, result): “tely generated H1,c-modules on which every P ∈ C[h∗]+ acts locally nilpotently. 5.4.. It is possible to reduce the study of Ob c to the study of Oc thanks to recent work of Bezrukavnikov and Etingof, [15]. Given λ ∈h∗/W we let Gλ be the G-stabiliser of any lift of λ to h∗: this is of course only wel”
• Completions of symplectic reflection algebras (2010, methodology): “7.1, we see that •♥,b is right adjoint to • ♥,b. Therefore • ♥,b= •†,b. 5.6. Induction and restriction functors and the annihilators. The main application of the isomorphism of completions theorem in [BE] is the construction of functors between the categories Oof appropriate Cherednik algebras. Name”
• Categorifications and cyclotomic rational double affine Hecke algebras (2013, background): “Bezrukavnikov and Etingof [5] defined parabolic induction and restriction functors OIndWW 1 : OcpW 1, h{hW 1 qR Ñ OcpW, hqR, OResWW 1 : OcpW, hqR Ñ OcpW 1, h{hW 1 qR.”
• Quantum toroidal algebras and their representations (2008, background): “ do not have a Chevalley presentation for these algebras, this theory should rely on Drinfeld generators. - Bezrukavnikov and Etingof defined induction and restriction functors for Cherednikalgebrasin [BE]. As suggestedby P. Etingof, it would be veryinteresting to relate it to the fusion product of q”
• Representations of Rational Cherednik algebras with minimal support and torus knots (2013, methodology): “l often drop the subscript n from the notation for these idempotents. REPRESENTATIONS WITH MINIMAL SUPPORT AND TORUS KNOTS 5 be explicitly computed from the construction of [CEE], and it follows from [BE] that this support is minimal). This means that the annihilator of Lm n (n0λ) is the maximal ide”
• Heisenberg algebras and rational double affine Hecke algebras (2010, unknown): “The parabolic induction/restriction functor relatively to the point b is a functor [1] Indb : O(Wb, h/h b) → O(W, h), Resb : O(W, h) → O(Wb, h/h b).”
• Isomorphisms of quantizations via quantization of resolutions (2010, background): “Namely, let eb be a point in the preimage of b in V and set G:= Geb. Then define the algebra H as the quotient of TV[c]#G by the relations (6.5) [u,v] = hω(u,v)+ Xr i=1 ci X s∈Si∩G ωs(x,y)s, Following [BE], consider the centralizer algebra Z(G,G,H∧0) that comes equipped with an embedding CG→Z(G,G,H∧0”
• Generalized Jack polynomials and the representation theory of rational Cherednik algebras (2010, methodology): “On the other hand, we can show that all such parameters are aspherical using Theorem 7.5 of [10] and the results of [ 2 ].”
• On category O for cyclotomic rational Cherednik algebras (2011, background): “Cherednik algebra, category O, derived category, deformation quantization, categorification”